white matter microstructural property decoding from gradient echo data using realistic ... ·...

1

White Matter microstructural property decodingfrom gradient echo data using realistic white matter

modelsRenaud Hedouin∗, Riccardo Metere∗, Kwok-Shing Chan∗, Christian Licht‡, Jeroen Mollink†, Anne-Marie

van Cappellen van Walsum†, Jose P.Marques∗

∗ Radboud University, Donders Institute for Brain, Cognition and Behaviour6525 HR Nijmegen, Netherlands

† Radboudumc, Medical Imaging and Anatomy, Nijmegen, Netherlands‡ Computer Assisted Clinical Medicine, Medical Faculty Mannheim, Heidelberg University, Germany

Index Terms—White matter models, Microstruc-tural properties, Magnetic susceptibility, Deep learningnetwork

Abstract—The multi-echo gradient echo (ME-GRE)magnetic resonance signal evolution in white matterhas a strong dependence on the orientation of myeli-nated axons in respect to the main static field. Althoughanalytical solutions, based on the Hollow CylinderModel have been able to predict some of the behaviourthe hollow cylinder model, it has been shown that real-istic models of white matter offer a better descriptionof the signal behaviout observed.

In this work, we present a pipeline to (i) generaterealistic 2D white matter models with its microstruc-ture based on real axon but with arbitrary fiber volumefraction (FVF) and g-ratio. We (ii) simulate their in-teraction with the static magnetic field to be able tosimulate their MR signal. For the first time, we (iii)demonstrate that realistic 2D models can be used tosimulate an MR signal that provides a good approx-imation of the signal obtained from a real 3D whitematter model obtained using electron microscopy. Wethen (iv) demonstrate in silico that 2D WM modelscan be used to predict microstructural parametersin a robust way if multi-echo multi-orientation datais available and the main fiber orientation in eachpixel is known using DTI. A Deep Learning Networkwas trained and characterized in its ability to recoverthe desired microstructural parameters such as FVF,g-ratio, free and bound water transverse relaxationand magnetic susceptibility. Finally, the network wastrained to recover these micro-structural parametersfrom an ex-vivo dataset acquired in 9-orientations inrespect to the magnetic field and 12 echo times. Wedemonstrate that this is an overdetermined problemand that as few as 3 orientations can already providecomparable results for some of the decoded metrics.

[Highlights] - A pipeline to generate realistic whitematter models of arbitrary fiber volume fraction andg-ratio is presented; - We present a methodology tosimulated the gradient echo signal from segmented2D and 3D models of white matter, which takes intoaccount the interaction of the static magnetic fieldwith the anisotropic susceptibility of the myelin phos-pholipids; - Deep Learning Networks can be used todecode microstructural white matter parameters fromthe signal of multi-echo multi-orientation data;

I. Introduction

White matter (WM) consist mainly of myelinated axonsand plays an important role for the transmission of infor-mation across the brain. The myelin sheath surroundingthe axons acts as an electric insulator, thus increasingthe transmission speed of the pulses. The development ofmyelin played a key role in evolution and the apparitionof large vertebrate [1] and it is still central in brain mat-uration. The degradation of myelin, commonly referred toas demyelination, is present in various neurodegenerativediseases, and leads to severe motor and mental disabil-ities [2]. Such neurodegenerative disorders (e.g multiplesclerosis) show high variability among individuals, and itis difficult to predict and understand the course of thedisease by only counting the number of lesions or com-paring the values obtained in magnetic resonance (MR)relaxometry [3]. Therefore, non-invasive imaging methodsthat can investigate the WM microstructure and measuremyelin properties may offer important means of studyingneurodegenerative diseases, providing crucial informationfor diagnosis, monitoring progression and assessment ofpotential treatment effectiveness.

Direct MR imaging of the myelin is challenging due tothe ultra-short transverse relaxation time of the phospho-lipid proton (T ∗

2 = 0.3 ms). Nevertheless, several attemptshave been performed using zero or ultra short echo timetechniques [4], [5]. Alternatively, myelin can be probedindirectly using magnetization transfer techniques [6], [7],multi-echo spin- or gradient- echo sequences [8]. However,the detection of myelin water remains challenging due toits short T2 and T ∗

2 values (∼10 ms). In this paper, we willfocus on myelin water imaging using a multi-echo gradientecho (ME-GRE) sequence.

WM is a complex environment composed not only ofaxons but also different types of glial cells, vessels andmore. However, the biophysical models typically used inmagnetic resonance imaging (MRI) is simplified to 3 com-partments: intra-axonal, myelin and extra-axonal waterprotons. Axons in WM have various shapes and sizes, with

(which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. The copyright holder for this preprintthis version posted June 24, 2020. . https://doi.org/10.1101/2020.06.23.127258doi: bioRxiv preprint

https://doi.org/10.1101/2020.06.23.127258

2

a diameter ranging from 0.1 µm to 2 µm for unmyelinatedaxons and from 1 µm up to 10 µm for myelinated axons[9], but are typically modeled as cylinders. The myelinsheath, formed in the central nervous system (CNS) byoligodendrocytes represents approximately 80% of thebrain’s dry weight, consists of tightly packed phospholipidbi-layers united by the hydrophobic tails, separated bywater layers [10]. These phospholipids because of theirelongated form and their radial organisation around theaxon have an anisotropic magnetic susceptibility [11],[12] which is diamagnetic when compared to surroundingwater. These microstructural features are believed to bewell approximated by a tensor with cylindrical symmetry.Thus, the susceptibility of myelin can be written as thesum of an isotropic (Xi) and anisotropic (Xa) component:

X = Xi +Xa = χi

1 0 00 1 00 0 1

+ χa

1 0 00 −1/2 00 0 −1/2

(1)

where χi and χa are scalar isotropic and anisotropic sus-ceptibility multiplicative constants, respectively. Variousvalues have been reported in the literature of myelin forχi ranging from −0.13 to −0.06 ppm and χa ranging from−0.15 to −0.09 ppm [13]–[15] (with ppm considered withrespect to the magnetic susceptibility of pure water).

In the presence of a strong magnetic field, a secondarymicroscopic magnetic field perturbation is created by thesephospholipids [16]. This secondary field can be observedin both magnitude and phase of a multi-echo GRE signal[17]. One manifestation of the anisotropic magnetic suscep-tibility of myelin is that the MR signal of a GRE sequenceshows a dependence on the orientation of the fibers relativeto the main magnetic field. For example, it has beenshown that simple T ∗

2 maps are orientation dependent[18], and hence unsuitable for the estimation of myelinproperties. Part of this orientation dependence can beaccounted for using a priori knowledge of fiber orientations[19]. More advanced biophysical modeling of the signaldecay allows for the estimation of orientation dependentand independent components [20]. One alternative way toexplore this complex interaction between magnetic fieldand the magnetic susceptibility of myelin, is myelin waterimaging (MWI). In MWI the MR signal is fitted usinga 3-pool model [21], [22], with each compartment havinga specific relaxation time and frequency offset, yet thisapproach suffers from over-fitting issues [23], [24]. Usingthe hollow cylinder model (HCM) [25] and fiber orientationinformation derived from DWI, our group has shown thatit is possible to overcome some of the ill-posedness natureof MWI using what was named Diffusion Informed MyelinWater imaging (DIMWI) [26]. Yet it has been previouslyshown that more complex and realistic WM models basedon electron microscopy data [11] are better able to charac-terize the signal ME-GRE signal than the more simplisticHCM.

In this work, we present a novel approach to map some

of the properties of WM microstructure by modeling thebehavior of the MR signal of a ME-GRE imaging sequencemeasured for multiple orientation of the tissue sampleand hence of the WM fiber bundles. For this, we firstdevelop a method to generate a hypothetical 2D WMmodels based on realistic axon shapes. These models arethen used to simulate the ME-GRE signal for differentaxon and myelin properties (notably, their relative size andvolume fractions) and validated by comparing them to real3D WM models based on electron microscopy data. Thesesimulations are used to construct a dictionary of the ME-GRE, covering a wide range of WM properties. Finally thedictionary was used to train a deep neural network to mapME-GRE acquired signal using multiple orientations of asample in respect to the magnetic field into white matterproperties. We tested this deep learning method in variousscenarios both in silico and in vitro.

II. Methods

A. 2D WM model

The magnetic susceptibility of myelin relative to itssurrounding creates a magnetic field, that although small,affects the MRI signal both in phase and magnitude. Thisphenomenon have been used in the past to study WM ori-entation [19], [25] and can be studied both analytically andnumerically considering various simplified WM models.

1) Hollow cylinder model: The HCM, proposed byWharton and Bowtell, is commonly used to approximateWM microstructure [14]. The myelin sheath is representedby an infinite hollow cylinder with an inner radius ri andan outer radius ro. The inner part of the hollow cylinderis the intra-axonal compartment and the external part isreferred as the extra-axonal compartment.

This cylindrical representation of WM into 3 compart-ments allows an analytical derivation of the field pertur-bation in each of those regions and characterization WMusing:

• Fiber volume fraction (FVF) - the proportion ofmyelinated axon within the model

• g-ratio - the ratio between the intra-axonal radius (ri)and the myelinated axon radius (r0):

g-ratio =riro

(2)

This solution, very convenient to model, offers for examplean analytical estimation of the fiber-orientation depen-dence of R2∗(1/T2∗) map [25].

However, it has been recently demonstrated that theHCM has intrinsic biases compared to a more realisticWM model created from electron microscopy data [11].The circular axon shapes create artificially large frequencypeaks, in particular within the intra-axonal compartment,which are not present in a realistic model. In the followingsection we will present the creation of a realistic 2DWM model based on real axon shapes and realistic sizedistributions.


https://doi.org/10.1101/2020.06.23.127258

3

2) Electron microscopy based models: In this study, weused an 2D electron microscopy image of an entire sliceof a canine spinal cord from an histology open database1 as our database of axon shapes. The sample is 5mmwidth and 7.5mm long with a 0.25µm resolution whichcorresponds to a 20.000× 30.000 image. An open-sourcesegmentation software was used to segment the image thatlead to a collection of ∼ 600.000 myelinated axon shapes[27]. The resolution is sufficient because we do not want tosegment unmyelinated axons that have been shown to haveno significant impact in the obtained ME-GRE signal [11].The unmyelinated axons are therefore included within theextra-axonal space. In case of a realistic axon shape, theg-ratio is redefined as the square root of the ratio betweenthe intra-axonal surface and the outer surface (measuredas the number of myelinated pixels with at least one sidein direct contact with intra or extra-axonal space).

3) Axon packing algorithm: A set 400 of axon shapeswere randomly picked from the collection above to createa realistic 2D WM model with predefined FVF and g-ratio.To do so, we developed an axon packing algorithm basedon an existing software [28] that had been initially devel-oped for cylindrical axon models. The packing process isperformed as follow (see Fig 1):

Algorithm 1: Axon packing

Data: Set of N myelinated axon shapesInitialization: N axons equally spaced on a gridcurrent FVF = initial FVFwhile current FVF < maximum FVF do

Axons are attracted to the grid centerAxons which overlap repulse each othercurrent FVF = FVF within a mask

end

In the current implementation, as the axon shapes arepicked randomly, they do not necessarily fit optimallytogether (during the attraction and repulsion process,the axon is not allowed to rotate) which creates smallgaps within the model. The maximum FVF parameter,corresponding to a model where the axons are highlypacked while avoiding overlap was empirically found to be0.85. According to literature, such an FVF value alreadyrepresents a WM model with a very high axon density [29].

4) Obtaining an expected FVF: Once the maximumFVF for a given collection of axons is achieved, thispacked WM model was used to obtain a new model withan a different FVF. Two different methods, illustratedFig 1, were proposed: (i) randomly remove axons or (ii)spread the axons from the figure center. The first methodcreates important gaps within the extra-axonal space thatcould correspond to glial cells or bundles of unmyelinatedaxons, while the second method creates a more uniformlydistributed WM model. Based on the EM data visuallyexplored up to now, both could be valid representations.

1https://osf.io/sgbm8/

Their corresponding field perturbation histograms wereclose enough and both models were used to enforce thediversity of our WM model dictionaries.

Fig. 1. Top row: 400 axons are placed on a grid (a) and packedfollowing an attraction/repulsion method (b) until to reach high FVF(c). Bottom row: Zoom on the mask delineated by the red square. Adesired FVF is reached spreading the axons from the center (d) orrandomly removing some axons (e). Keeping the same axons andthus the same FVF, the myelin thickness can be modified to obtainan expected g-ratio (f)

5) Change the g-ratio: Finally, the mean g-ratio of themodel was modified, while keeping the FVF constant. Thisoperation was performed on an axon-by-axon basis bydilating or eroding the inner myelin sheath by one pixeldepending on whether the g-ratio was to be decreasedor increased. Each axon has a given probability to berandomly picked, this probability is linked to its diameter.As the dilatation/erosion is fixed to one pixel, largeraxons need to be picked more frequently to respect theoriginal proportion of FVF. The modification of the g-ratio is illustrated in Fig 1. Eventually, different modelswith similar FVF and g-ratio can be created using ourlarge axon shapes database and the code made availablein the toolbox.

B. Signal creation

To be able to use these 2D models to simulate the ME-GRE signal, we need to define the susceptibility of pixelelement, compute the induced magnetic field perturbationand eventually simulate the signal evolution in this inho-mogeneous environment.

1) Magnetic susceptibilities: The susceptibility tensorXR within the myelin sheath in a 2D model is determinedby the phospholipid orientations φ on that plane:

XR = Rz(φ) ·X ·Rz(φ) = Xi +Rz(φ) ·Xa ·Rz(φ) (3)

with Rz(φ) the 3D rotation matrix around the z axis.In simple cases, as for the HCM, the computation of φis trivial. However, for more complex axon shapes, thereis no proper definition of a radial orientation betweentwo boundaries. The orientation of the phospholipids is


https://doi.org/10.1101/2020.06.23.127258

4

estimated on an axon-by-axon basis. First, the selectedaxon is placed in a small matrix (including 10 pixelsof each side of the axon edges for computational timeconsiderations), then the extra-axonal, myelin and intra-axonal compartments are given the values of 0,1 and 2respectively. The resulting map is smoothed with a 2DGaussian filter with a width of 5× 5 to create a smoothedpyramidal structure, if the myelin sheath is too large andstill contains piecewise constant part after smoothing, theprocess is repeated, and finally a gradient direction mapis computed. As the map is smoothly varying from 0 to2 within the myelin compartment, the gradient at eachpoint will define the steepest direction from the extra- tothe intra-axonal space and should follow the phospholipidorientation (see Fig 2).

2) Field perturbation: From the phospholipid orienta-tion map, the susceptibility tensor map can be calculatedusing Eq. 3. The susceptibility tensor map is used tocompute the field perturbation in the frequency domain asdescribed in [30]. An illustration of the field perturbationgenerated by a single axon with several B0 orientations isshown in Fig 2. The induced field perturbation stronglydepends on the B0 orientation. A magnetic field parallelto the axon orientation only creates a small negativefield shift within the myelin sheath while a perpendicularmagnetic field creates much stronger perturbations withinthe 3 compartments. The overlapping frequency spectra ofthe 3 compartments make them hard to disentangle.

Fig. 2. First row: Phospholipid orientation estimation. (a): Originalaxon, the extra-axonal part is filled with 0, the myelin with 1 andthe intra-axonal with 2. (b): Model smoothed with a Gaussian filter.Right : (c) Gradient orientation computed on the smoothed map.Second row: Field perturbation for one axon with 3 different magneticfield orientations. Third row: Corresponding histograms computedwithin the red square to keep a reasonable FVF.

3) ME-GRE signal: ME-GRE signals are computed as:

TABLE IMiddle column: Parameter range used in our dictionary.Third column: Usual parameter values found in WM. *The

relative water weight depend on the acquisitionparameters, flip angle and TR, subsequently there is nousual value and the one presented is used within the

typical WM deep learning experiment.

Modelparameters

DictionaryTypical WM

valuesFVF 0.1:0.1:0.8 0.7 a

g-ratio 0.5:0.05:0.85 0.65 b

χi -0.2:0.1:0.2 -0.1 c

χa -0.1 (fixed) -0.1 a

T2,Intra−Extra 20:20:100 60 d

T2,Myelin 4:4:20 16 d

wMyelin/wIntraExtra 0.5:0.5:3 2*Fiber orientations 20 /

a (Choy et al., 2020) [31]b (Mohammadi et al., 2015) [32]c (Wharton et al., 2012) [14]d (Xu et al., 2018) [11]

S(t) =3∑

n=1

(wn exp

(−tT2,n

)∑r

exp (−itγ∆Bn(r))

)(4)

where each compartment has a specific transverse re-laxation T2,n, a water weight wn reflecting the watersignal, which includes proton density and T1 saturationeffects, and a corresponding field perturbation ∆Bn(r). Anillustration of the ME-GRE signals simulated using Eq 4is shown with two examples of WM geometry in Fig 3.

MRI data amplitude depends, not only on the magneti-zation amplitude, but also on the RF coil sensitivity andreceiver gain. The phase depends on the RF transceiverand on the quality of the B0 shimming and presence offields due to the susceptibility of neighbouring pixels. Tobe able to compare our signal simulations to real data, thissignal (and the real data) is normalized as follow:

|S(t)| = |S(t)|/|S(1)| (5)

arg(S(t)) = arg(S(t))− p1 − p2 × t (6)

where arg(S(t)) is the phase of the signal and p1 andp2 are the coefficients of p(t) = p1 + p2 × t, the line thatbest fits the original phase arg(S(t)). This normalizationis done to keep the second order evolution while removingthe linear part that corresponds to the phase and fre-quency offset of the MRI acquisition which are hardware-dependent.

4) Model validation: While the realistic 2D WM modelshave been shown to better represent the ME-GRE signalof WM than the simple HCM, they assume the replicationof the same structure along the third dimension resultingin bundles that are unrealistically aligned and cannot rep-resent the natural dispersion present in a real axon bundle.Dispersion can occur not only in regions of fiber crossing,


https://doi.org/10.1101/2020.06.23.127258

5

Fig. 3. WM models with different FVF and g-ratio and their corresponding field perturbations. The ME-GRE signal magnitude and phaseare presented for 4 difference χi values. The other parameters are fixed according to literature values (see table I.

fiber kissing, but also in regions traditionally expected tobe unidirectional such as the corpus callosum [33]. How-ever, 3D models are hard to construct, not only becauseof the lack of 3D EM data (that could represent a groundtruth), but also because of the complexity of 3D axonpacking [34]. Also, it could be the case that the 2D axonshapes, used in our realistic WM modeling, are elongateddue to being obtained from cutting through axons thatwere not perpendicular to the surface. Furthermore, in thecase of our application, the estimation of the susceptibilitytensor map and the field perturbation in 3D models wouldmake the process even more time consuming. We havedesigned a small study, presented in the Appendix , toevaluate the ability of our 2D models to represent a real3D model with comparable microstructural properties.

C. Dictionary creation

A dictionary of signal evolution can be created usingthe simulated ME-GRE signals in the presence of differentWM model. Such dictionary can be used to derive themicrostructural tissue properties from the ME-GRE signalby root mean square minimization, as done for example infingerprinting [35]. Alternatively, a deep learning networkcan be trained to learn the tissue properties from thedictionary as will be demonstrated later.

The WM model and the magnetic field distributionspresent on each of its compartments depends on 5 mi-crostructure related parameters: FVF, g-ratio, χi andχa, as well as the fiber orientation. For the purpose oftraining a deep learning network, we considered repeatingsimulations with various axon packing using the sameproperties as aforementioned. The ME-GRE signal fromeach WM model depends on the specific NMR proper-ties of each compartment (wn, T2,n). This would resultin 6 supplementary parameters. To minimize dictionary

size, the T2s and water weights of the intra-axonal andextra-axonal compartments were defined equals and thesignal amplitudes was always normalized (so that

∑wn =

1), reducing the number of parameters from 6 to 3:w = S0,IntraExtra/S0,Myelin referred as the relative waterweight, T2,Myelin, T2,Intra−Extra. The parameter ranges,used to construct the dictionary, are presented in TableI along with typical WM values. The dictionary has 8dimensions, with 5 to 20 entries per dimension leadingto 7.680.000 vectors. In the following in silico and exvivo experiments, all the dictionaries have those sameparameter ranges.

Each entry of the dictionary is composed by the normal-ized signal magnitude and phase (or real and imaginarycomponents, 2 x nTE with nTE the number of echo timesin the simulation) and an additional entry encoding thefiber orientation information characterized by the anglebetween the fiber and the static magnetic field. When de-riving the microstructural properties from measurementswith multiple orientations with respect to the magneticfield, the signal is concatenated along the n orientationswhich leads to a vector size of n ·(2TE+1). An illustrationof such simulated normalized signals magnitude and phasewith different orientations is presented Fig 4. Converselyto a single orientation dictionary, this multi-orientationdictionary is only valid for a specific set of rotations usedin a specific acquisition.

D. Deep Learning

The ME-GRE signal dictionary was used to train a deeplearning network using Keras [36]. For all the followingexperiments, the dictionaries were trained on 7 entiresets of WM models and evaluated by the loss functionon another set of WM models, which correspond to avalidation split of 0.125. This network is composed with 3


https://doi.org/10.1101/2020.06.23.127258

6

Fig. 4. The ME-GRE signal is simulated with 6 magnetic fieldorientations, θ = 0−π/2 equally space (see arrows), for WM modelswith different FVF from 0.1 to 0.8 (4 models for each FVF). The topand bottom row represents respectively the signal real and imaginarypart for each 6 orientations separated by a vertical black line.

hidden layers of size 2∗ li ∗ lo, 1.5∗ li ∗ lo, 1.25∗ li ∗ lo, li andlo being the concatenate signal length and the number ofparameters, with a respective drop out of 0.4, 0.2, 0.1 usingan tanh activation function and an additional linear layer.Both inputs and outputs were normalized, a stochasticgradient descent optimizer was used and the loss functionis a mean absolute error.

To gain experience on the ability and limitations of ournetwork to derive microstructure properties, its perfor-mance was first tested on simulated data. Particularly wewanted to evaluate what was the optimum echo time rangeand the number of echoes, as well as study the gains asso-ciated with different numbers of sample rotations neededto successfully recover WM properties (which will affectour data acquisition protocol). The design and training ofthe network was also subject of careful attention. The deeplearning hyperparameters were tuned following an empir-ical approach, with the chosen ones giving both accurateresults and robust to the change of signal parameters.

The validation loss function (mean absolute error of theparameters estimated on a validation data set - one set ofWM models which is not used for training) was used as ametric to evaluate the convergence of the network. All theparameters, within their range, were re-scaled between 0and 1, to make validation loss a less arbitrary number.This metric is an average of the mean absolute errorfor each parameter, thus, it does not allow to make finecomparisons. Despite this remark, the validation loss isa classic and robust way to evaluate the training processwith an unique number.

1) Deep Learning performance evaluation on simulateddata: The robustness of the parameter recovery was testedby adding a complex white noise (0 %, 0.5 %, 1 %, 2 %and 4 %) to a ME-GRE signal on a dictionary used in thetraining and validation processes. The first 3 columns ofTable II summarize the parameters used in the creation ofthe dictionary and training of the network. The rotationsused were chosen to mimic the experimental protocol usedon an ex vivo acquisition described later in this section.The noise levels mentioned above are relative to the signal

amplitude at the first echo, TE=2.15 ms

The ME-GRE signal of a given white matter modeldepends on the magnetic field orientation in respect toits structure (see Fig 4), this lead us to adopt a multi-orientations approach when trying to decode WM mi-crostructure properties. However, as an increased num-ber of orientations means a longer acquisition time, weperformed a theoretical comparison study to estimate thebenefit of using a large number of orientations vs a reducednumber of orientations with higher SNR. A dictionary with16 optimal orientations was created for 3 different noiselevels (0,1 and 2%). In order to maximize information,each fiber should have the largest possible range of θ from0 to π/2. To do so, the 16 3D rotations had evenly spreadaxis on the sphere with a common π/2 angle. Then, for arange of number of orientations from 1 to 16, a subset ofthis dictionary was used to train a deep learning network.

The influence of the number of echoes on the deeplearning parameter recovery performance was tested. Todo so, several networks were trained with a fixed echospacing (3.05 ms - mimicking our experimental protocol),a various number of TE (5, 10, 15, 20, 25 and 30) andnoise levels. At this stage no considerations of the impacton T1 weighting were included on the analysis.

Finally, we tested the deep learning for one set of real-istic parameter values of WM (see Table II), that allowsto detail the behavior of each parameter individually. Thesignal was simulated 125 times for 8 independent whitematter models leading to 1000 signal simulations with eachdifferent noise level. We tested two methods to recoverthe parameters: (i) using a deep learning network trainedwith a noise matching to the simulated noise; using a deeplearning trained with a maximum noise level regardless ofthe simulated signal noise.

E. Ex vivo data acquisition

A formalin fixed post-mortem brain (female, 88 yearsold, 26 hours of post-mortem interval and 7-month fixationperiod) was scanned in a 3T scanner (Siemens, Prismafit).The brain was scanned in 9 orientations relative to thestatic magnetic field. To avoid brain deformation betweendifferent rotations, a customised 3D brain holder was builtand used throughout the scanning session, see Fig. 5. Priorto scanning, formalin was washed out in distilled waterand prepared in low pressure environment, using a vacuumpump at 20mBar during 12h to remove all air bubblestrapped in the various cortical sulci. During this periodthe brain was occasionally rotated to ensure removal ofair trapped inside the ventricles.

Fig. 5. Brain holder: (a) in its sphere, (b) alone, (c) open


https://doi.org/10.1101/2020.06.23.127258

7

TABLE IIThis table describes the dictionary parameters, (TEs, rotations, noise level, number of models) and a deep learning

parameter (number of epochs) associated with each experiment. Four first columns: in silico experiment, Last column:ex vivoexperiment.

ParameterExperiment Epochs

dependenceTEs dependence

Rotationdependence

Typical WM Ex vivo data

TEs 2.15-3.05-35.7 1.8-3.2-14.6/94.6 2.4-4.4-50.7 2.15-3.05-35.7 2.15-3.05-35.7Rotations 9 6 1 to 16 9 9Noise level 0, 0.5, 1, 2, 4% 0, 1, 2% 0, 1, 2% 0, 0.5, 1, 2, 4% 4%

Number of models 8 8 8 8 8Epochs 40 20 40 40 40

For each head positions the following protocol wasrepeated: - (a) 3D monopolar ME-GRE with 12 echos(TE = 1.7 : 3.05 : 35.25ms, TR = 38 ms), with a1.8mm isotropic resolution and Matrix size (128x128x128),acquisition time 8.21 mins. This protocol was repeated 6times with flip angles α = 5°/10°/15°/20°/35°/65°); - (b)an MP2RAGE with 1mm isotropic resolution was acquiredfor co-registration purposes. The MP2RAGE parameterswere adapted to be able to map the short T1 values presentin fixed tissue (TR/TI1/TI2=3s/0.311s/1.6s α1/α2); Fi-nally, for the last sample position, DWI protocol was ac-quired to provide fiber orientation information (TR/TE =3.78s/71.2ms, 256 diffusion-encoding gradient directions,b = 2500s/mm2). Because the formalin fixation processand the reduced temperature of the sample compared toin vivo (Room Temperature ' 23°) significantly reducewater diffusivity, the protocol was repeated 20 times toachieve robust fiber orientation information.

F. Ex vivo data processing

Each of the 9 MP2RAGE images from the 9 brainrotations were co-registered to a reference position us-ing FLIRT from fsl [37]. Corresponding transformationswere then applied to the ME-GRE data (magnitude andunwrapped phase separately), finally the registered datawere normalized following Eq 6. A DTI was estimated foreach DWI and the 20 DTIs were averaged using a log-Euclidean framework [38]. Eventually, the fiber orientationwas defined as the main orientation of the average tensor.

A ME-GRE dictionary was simulated for this particularacquisition, and the corresponding deep learning networkwas trained using the parameter ranges described in TableI and II. Finally, the microstructure parameter maps(FVF, g-ratio, χi, T2,Myelin . T2, Intra − Extra, andthe relative water weight) were estimated individually foreach set of flip angles. This resulted in 6 independentsets parameter maps, where only the relative water weightterm is expected to vary across acquisitions. It was thuspossible to compute the mean and standard deviation ofthe microstructure parameter maps that were expected toremain constant across flip angles to estimate the precisionof those measurements.

Finally, the last experiment was performed by using a re-stricted number of rotations that can be achived during anin vivo experiment. Among the 84 possible combinations

of 3 rotations chosen within the original 9 rotations, the 10ones that insured the largest fiber orientations ranges werepicked. The subsets of ex vivo data for the 10 combinationsof 3 rotations with a flip angle of 20°, the correspondingdictionaries, and deep learning networks were created,leading to 10 entire sets of brain parameter maps. This wasused to compute the mean and standard deviation acrossdifferent combinations of 3 rotations. Finally, the absolutedifference maps between the mean parameter maps with 3rotations and the original ones with 9 rotations, both witha flip angle of 20°, were estimated.

III. Results

A. Deep learning performance on simulated data

1) Noise level: Figure 6(a) shows the dependence of theloss function of the deep learning network for 5 differentnoise levels as a function of the number of epochs used.After a fast drop during the first 3-5 epochs, the lossfunction continues a slow decay, reaching a plateau for thenoisier signals. Interestingly, the loss functions on the testdata (solid lines) are slightly lower than on validation data(dashed line). This difference is attributed to the fact thatthe validation loss function is averaged along the entireepoch, whilst the test loss function is computed at theend of each epoch. From this analysis we concluded that20 epochs should be a good compromised between trainingefficiency and parameter recovery.

2) Echo times: Fig 6(b) presents the dependence of theloss function on the number of echo times used. It showsthat the wider the range of echo times the lower the lossfunction is. The loss function clearly improves between5 to 15 TE (49ms), but its improvement is smaller afterthat, even if a plateau has never been totally reached forsignal with noise even after 30 echos. Our simulations didnot include any echo time dependent noise, arising fromphysiological noise or scanner drifts, which are commonin gradient echo acquisitions, and would make later echotimes less useful for decoding. We postulated that 20 echoswould be sufficient for an experimental protocol.

3) Number of magnetic field orientations: Fig 6(c)shows that, as expected, the loss functions decreases whenincreasing the number of rotations for all noise levels.Note that in interest of computation time, the subset ofrotations might not be optimal for all number of rotations


https://doi.org/10.1101/2020.06.23.127258

8

(a) (b) (c)

Fig. 6. Deep learning training evolution for different noise levels relative to several acquisition parameters. The solid line is the loss functionwhilst the dashed line is the validation loss function, that represents the same mean absolute error respectively computed on the train andon the test data set. (a) Training along the number of epochs. (b) Training along the number of echoes. (c): Training along the number ofrotations.

tested (as a subset of the initial 16 orientations wasused). Furthermore the specific number/set of rotationsdepends on the orientation of the fiber of interest. Thedeep learning benefits from the first 3-6 distinct rotations,similar to what has been demonstrated for Susceptibilitytensor imaging [39] and for fiber orientation mapping [14],and plateaus after this. In a given acquisition time we caneither decide to have an improved SNR per orientationor increased number of rotations. When moving from 1to 2%SNR levels this corresponds to an decrease of theacquisition time by a factor 4 or number of rotations. Thus16 orientations at 2% noise could be acquired in the sametime as 4 orientations at 1% noise level. It can thereforebe concluded that there is a benefit in maximizing thenumber of orientations beyond 5 as the loss functionfor 16 rotations at 2% noise was the same as that of 6orientations and 1% noise. In our acquisitions, we used 9to 10 orientations, to avoid excessive acceleration of eachorientation, as this could bring parallel imaging artifactsinto play when trying to further reduce the acquisition perorientation.

4) Selective set of parameters: Fig 7 shows the perfor-mance of the deep learning networks to recover the variousmicrostructural parameters of what could be considered atypical white matter model. Although the average recov-ered parameters are closed to the original ones regardlessthe signal noise level many of the differences would bestatistically significant. Particularly, the relative waterweight suffers a constant positive bias for all networks andsimulated signal. Surprisingly, the standard deviation forall parameters (excluding χ and T2,Myelin ) is considerablylower when the deep learning was trained with a 4%noise level rather then the matched noise level. Thus, adictionary with a high noise level was used in our ex vivoexperiment presented in the following. When comparingthe width of the various distributions, compared to therange used in the training the network (see Table I), thevalued of χ, g-ratio and relative water weight are likely tohave the largest biases and noise.

B. Ex vivo experiment

The 6 microstructure parameter maps obtained from theex vivo brain with a flip angle of 35 are presented in Fig 8,along reference images for coronal and sagittal views from

the ME-GRE and MP2RAGE for visual comparison withmore standard contrasts. The microstructure parameterswere computed with a network with a 4% noise level.White matter is clearly discernible from grey matter anddeep gray matter on the FVF and relative water weightmaps. It should also be noted that FVF and the intra andextra axonal T2 have a very strong contrast between whitematter and deep gray matter (although the latter hasreduced contrast between grey and white matter). Thatobservation is particularly interesting because it suggeststhat with our modeling we were able to remove myelincontributions to the T2 contrast. On the other hand,the g-ratio map and T2 of myelin seem to have largecontrast within white matter which are expected to varywithin the brain. The sagittal maps show that an higherFVF, lower T2 of myelin and lower g-ratio in the corpuscallosum compared to the rest of the brain. Interestingly,CSF presents an almost null FVF along with a high T2intra/extra axonal, which is to be expected as there areno structures generating an anisotropic signal evolutionin this region. The χi values of myelin within WM areslightly positive where a negative value is expected, thiseffect could be due to the fixation process.

In addition to considering one particular flip angle, asin Fig. 8 , Figure 9 shows the mean and standard devi-ation across the 6 flip angles used. Most microstructureparameters should not depend on the flip angle, exceptthe relative water weight that is related to the protondensity as well as the TR and the flip angle. The meanparameter maps have globally the same characteristicsthan the ones presented previously with a flip angle 35, seeFig 9. The corresponding standard deviation maps reveallow values, showing a good robustness of the parameterrecovery for several acquisitions. Finally, the relative waterweight maps are shown in the same figure. As expected,an increase along the 6 flip angles is visible, in particularfor the 2 higher flip angles, 35 and 60. Reflecting the factthat the T1 of myelin water is significantly shorter thanthat of free water.

Fig 10 shows the brain parameter maps computed fromseveral data subsets each using a different combinationsof 3 rotations. The mean parameter maps highlight theexpected brain structures, such as CSF, deep gray matter,WM, GM. Yet, the contrast seems lower compare to the


https://doi.org/10.1101/2020.06.23.127258

9

Fig. 7. Each box represents the estimation of one parameter recovery for 5 different signal noise levels (0%, 0.5%, 1%, 2%, 4%). Within abox, the left side use a single deep learning trained with 4% noise regardless of the noise level while the right side use 5 deep learning, eachone trained with a noise equal to the signal level.

parameter maps obtained with 9 rotations, in particularwithin deep gray matter, as illustrated by the absolutedifference maps. The standard deviation maps, estimatedacross 10 combinations of 3 rotations, reveal very lowvalues. Thus, the process seems very robust when the samenumber of rotations is considered.

IV. Discussion

A. White matter models: promise and limitations

We introduced a pipeline to create a simple but realisticbiophysical model to simulate the MRI ME-GRE signal.These WM models contain real axonal shapes and ag-ratio variability similar to what is reported in tissuesamples (data not shown), and have varying levels of fibervolume fraction within themselves as a result from theaxon packing approach. Yet, some effects are explicitlyoverlooked: (1) diffusion within the compartments, (2)chemical exchange and (3) other sources of susceptibilityperturbations beyond the myelin sheath. Diffusion hasbeen demonstrated to have a minor effect for white mattermodels based on EM data [11] when compared to theHCM or simple cylindrical perturbers [40]. Chemical ex-change between myelin water and myelin protons resultsin frequency shift, and thus, can be accounted for byadding an exchange term in the HCM [25]. A recent workbased on Generalized Lorentzian Tensor Approach directlyconsiders the water layer within the myelin to probe thisexchange [41]. The size of this frequency offset term has

been reported to be of 0.02 ppm in the corpus callosum[14], but models have been proposed that would makethis offset depend on the number of myelin layers andtherefore vary throughout the brain and fibre bundles [42].Yet, chemical exchange has been demonstrated to have alarger impact when measuring the longitudinal relaxationin white matter, which is an aspect that, for the sake ofcomplexity, we have not included in our dictionaries. Theextra-axonal compartment currently includes everythingthat is found outside of the axon. More classes withspecific properties could be used, particularly: free water(CSF and interstitial spaces); blood vessels; bound-watercompartment (that represents the water bound to macro-molecules present in cell walls and organelles [43]), andiron accumulated in ferritin, amongst other. Blood vesselsoccupy a very small fraction of tissue volume (1-4% in WMand GM, but deoxygenated (venous) blood has a muchlarger susceptibility difference to free water than myelin)and tend to follow the orientation of white matter axonbundles [18]. This is expected to introduce some degreeof T ∗

2 anisotropy that would act as a confound in our exvivo experiment. Ferritin, which is known to be stronglyparamagnetic, can be found everywhere in the brain (withincreasing quantities found from WM, GM to deep graymatter where it can be found in large quantities [44]). Onour current implementation, iron is expected to be equallydistributed in the intra and extra-axonal space. As a resultferritin will be mapped as a reduction of the intra and


https://doi.org/10.1101/2020.06.23.127258

10

Fig. 8. Brain parameter maps estimated from ex vivo acquisition with flip angle 35 in a sagittal slice cutting the corpus callosum andtransverse to coronal slice cutting through the globus pallius. Additional T1 maps, estimated from MP2RAGE, and ME-GRE magnitudefirst echo provide structural information for comparison.


https://doi.org/10.1101/2020.06.23.127258

11

Fig. 9. Top line: Mean parameter maps averaged across the 6 flip angles. Middle line: Corresponding standard deviation maps. Bottom line:Relative water weight for the 6 flip angles.

Fig. 10. Top line: Mean parameter maps averaged across 10 combinations of a subset of 3 orientations selected among the 9 originalorientations. Bottom line: Corresponding standard deviation maps.


https://doi.org/10.1101/2020.06.23.127258

12

extra axonal T ∗2 and the isotropic magnetic susceptibility

attributed to the myelin compartment is effectively thedifference between the susceptibility of myelin and thefree water compartments where there might be ferritininclusions.

B. Dictionary and deep learning

Many of the simplifications used in our white mattermodels arise from the need to restrict the number ofparameters associated with our network. The size of adictionary, which in this study had 7 dimensions (seeTable I), is around 10 GB, moreover, an increase in thenumber of variables mapped by the network will result inan increased noise of the parameters estimated. We believewe have restricted the modeling to the most relevantparameters. In particular, we have considered FVF andg-ratio inherent to the model, as described previously theextra-axonal space can have various types of constituents,thus the extra-axonal T2 cannot be fixed. We choose to freeχi (allowing this to incorporate magnetic susceptibilityin the intra-extra axonal compartment) and to fix χa asthe major contribution to the magnetic field perturbationcomes from the isotropic susceptibility [45]. The compart-ment water weights were represented by a single variable,the relative water weight that includes the water protondensity as well as the degree of T1-weighting (and chemicalexchange) of each compartment. If the myelin sheath isconsidered having the same properties all other the brainthat allows to fix the myelin T2 and release the anisotropicsusceptibility χa which was reported ranging from −0.15to −0.09 ppm [15]. A potential direction for future work isto investigate different sets of parameters. For example,one could link the myelin water concentration to thesusceptibility of the myelin sheaths by taking into accountthat the magnetic susceptibility of the phospholipids andwater are both known.

Our deep learning network seems robust and system-atically converges for each dictionary associated to anexperiment with multiple orientations as illustrated in Fig6. However, extensive manual fine-tuning of the networkhyper-parameters was required to achieve this level ofagreement. A more systematic approach, while potentiallydesirable, would need an excessively long computationtime. In the future, it may be possible to do this, whenaccess to improved computational resources becomes morecommon. The in silico analysis (see Fig. 7) shows that adictionary trained with a higher noise level is more robustto noise amplification than a dictionary with matchednoise levels. This was attributed to the noise allowing tosmear our differences associated with the fact that our”realistic model” produce different signals (see Fig 4) andnone of them actually corresponds to the actual whitematter mapped. An interesting experiment would be toevaluate the performance of a dictionary including alldifferent noise levels, closer mimicking the signal found inthe brain where regions further away from the receivercoils are bound to have a lower SNR. It was observed that

the level of noise is within the range that differentiatesour 2D models from a real 3D white matter for a relativelarge range of dispersion values, which effectively makesour network more generalisable.

C. Ex vivo experiment

The human brain scanned on our ex vivo experimentwas fixed in formalin for 7 months prior to the experiment.It is well known that the microstructural tissue propertieschange throughout the fixation process, and the final prop-erties of the tissue depend on: the post mortem fixationdelay, the fixation time, the concentration of formalin andthe temperature history [46]–[48]. The T1 map presented inFig 8 shows particularly small values revealing a stronglyfixed tissues where water has a reduced mobility. Thiswas also clearly visible on the DWI imaging, the meanADC in white matter being 0.3 mm2.s−1 when a normalin vivo values is above 0.8 mm2.s−1 [49]. Fresh tissuesdo not present such parameter changes and could be analternative option. However, our current protocol takes8h without the DWI, in such time window using freshtissues, would not be sufficiently stable to assume constantmicrostructural properties over time [50]. Thus, it seemsnecessary to use fixed human brain, however, the fixationtime could be reduced to 6-10 weeks.

The approach presented in this work may find applica-tion in the imaging of myelin water with gradient-echo-based acquisitions [8], [51]. Traditionally, myelin waterimaging using gradient-echo-based experiments tries tofit 9 independent parameters: three independent signals(separate amplitude, decay rate and frequency shift) foreach of the three compartments (intra- and extra- axonalwater and myelin) to a ME-GRE signal. The main short-comings of this approach are that: the model is knownto be insufficient (even the simple HCM predicts morecomplex signal evolution than 3 overlapping exponentialsignal decays) [25] and the fitting procedure is poorlyconditioned. In this work we have shown with simulationsthat we may obtain acceptable results with as few as 3orientations (rather than the 9 explored in the ex vivoexperiment). This may be further improved by includingadditional diffusion information specific to the intra- andextra-axonal water fractions, given the improved fittingperformances obtained recently [52]. Another avenue re-cently explored is, multi-compartment relaxometry [26],which uses variable flip angle measurements of the ME-GRE signal leveraging the different T1s of the free waterand myelin water compartments to further improve fittingperformance. Such frameworks could benefit from realis-tic WM models, by concatenating along flip angles usedinstead of being concatenated along rotations. Such anapproach would allow to separate water concentration ineach compartment from their T1 weighting giving a morephysical meaning to the parameter here dubbed as relativewater weight.


https://doi.org/10.1101/2020.06.23.127258

13

D. Ground truth validation

The recovery of microstructural information from the exvivo scans using realistic WM models follows the generalexpectations for FVF, free water T2, g-ratio and relativewater weight. In future experiments, these measurementsshould be validated by an independent method. One pos-sible avenue is to perform histology on selected excisedsamples after the scan which could provide a ground-truth,several methods exist to perform such histology analysis.CLARITY is a method using optical 3D imaging combinedwith a tissue clearing method which can provides neurondensity, fiber orientation distribution and cell type classi-fication [53]. X-ray microscopy is an instrument that cangenerate an entire 3D view of the interior of otherwiseoptically opaque samples in a non-destructive way [54].3D transmission electron microscope (TEM) uses electronas a source of illumination that provides an excellentresolution, better than a classic light microscope [55].In preliminary work, not shown here, we replicated thefixation process as well as the scanning protocol in a pigbrain, from which small sections were extracted to perform3D TEM analysis. Significant degradation of the myelinsheath for a number of axons was observed, where themyelin sheath appeared unpacked. Such a tissue changewould result in a decrease of g-ratio, increase of myelin T2and proton density, as well as a decrease of χi in respectto the in vivo case. However, the entire procedure betweenthe brain extraction and the histology was long and couldbe itself responsible by such alterations.

V. Conclusion

In this paper, we developed an open toolbox 2 to gen-erate 2D white matter models with controlled microstruc-tural properties such as fiber density and variability inthe g-ratio using publicly available electron microscopydata. Such models are used to estimate the correspondingfield perturbation and derive the multi-echo gradient-echosignals. Although our WM models are limited to 2D, wehave demonstrated that they can be satisfactorily usedto simulate 3D structures with a relatively high rangeof dispersion. Finally, dictionaries of GRE signals for7 different parameters (compartment relaxation values,magnetic susceptibility of myelin, fiber volume fractionand g-ration) associated with white matter properties at asubvoxel level were created were created. This single acqui-sition dictionaries can then be combined depending on themultiple rotation strategy to create a better conditioneddecoding problem and train a deep learning network. Weperformed several tests to estimate the quality of the sub-voxel parameter recovery using our network, depending onthe number of sample rotations, echo times used and noiseadded to the library. Unsurprisingly we found that thenetwork performs better the more data is given as input,thus more rotations and more echoes, but that because ofthe variations between different white matter modelsit is

2https://github.com/rhedouin/Whist

important to train the network with a level of noise biggerthan that of the available data.

The network was demonstrated on an ex vivo exper-iment was performed using a multi-rotation acquisitionand provided FVF, g-ratio, T2 maps clearly revealing brainstructures such as the CSF, GM, WM, corpus callosum orglobus pallidus. The parameter values (exception for χi )follow the expected patterns and were robust for differentacquisition protocols.

VI. Aknowledgements

The authors would like to thank Professor KarlaMiller and Dr. Michiel Kleinnijenhuis for providingus the 3D electron microscopy data and its seg-mentation that was used in the Appendix. This re-search was supported by the Nederlandse Organisatievoor Wetenschappelijk Onderzoek (NWO), Grant/AwardFOMaARNaAR31/16PR1056that sponsored the positionsof Renaud HAl’douin and Kwok-shing Chan.The authorswould also like to acknowledge the fruitful discussions onthe topic of this research with Prof. David Norris.

Appendix

A. 3d WM model

To validate the ability of the developed 2D realisticmodels to describe the 3D structures that exist in awhite matter voxel, we compared the signal associatedto the 2D models to those of a real 3D WM sample.A segmented [56] 3D EM of the corpus callosum of amouse was used for this comparison. The resolution ofthe initial 3D EM dataset was of 7.3x7.3x50 nm, whichwas subsequently down sampled by a factor of 7 resultingin a quasi isotropic resolution 51x51x50 nm. The FOV ofthe segmented piece was 20 × 20 × 20 µm (representedon a matrix of 400 × 400 × 400). Using the segmentation3D EM data, the FVF and g-ratio were computed to be0, 51 and 0, 67 respectively. Additionally, because the 3Dmodel does not consist of infinitely long structures that areparallel, the fiber dispersion was computed with respectto the average fiber orientation [57], and found to be lowσ = 0.04. In addition to this original model, to study theimpact of higher dispersion, 60 axons within the 3D modelwere selected to create a fiber orientation dispersion ofσ = 0.4. A mask surrounding the selected axons was usedto ensure remaining microstructural parameters remainedequivalent to those of the the whole sample (FVF = 0, 51and g-ratio = 0, 67). The 3D signal was computed onlywithin the mask and selected axons.

The magnetic susceptibility tensor, XR, was calculatedwith respect to the orientation of the phospholipids insidethe myelin sheath, using a 3D variant of the processdescribed in the methods section. The obtained tensor mapwas then used to calculate the magnetic field perturbationsin 3D, ∆B0(X(r)), as described in [30]. Both this processesare straightforward extensions of the 2D case and theirimplementation is available in our toolbox.


https://doi.org/10.1101/2020.06.23.127258

14

Fig. 11. Raw 3D EM data and myelin segmentation of size400x400x400. Frequency histogram of the computed axon orienta-tions present in the EM model and the average orientation, ~µ

B. Comparison between 2D and 3D field perturbations

To simulate the fiber dispersion within the 3D samples,an artificial dispersion was introduced into the 2D modelsby computing the field perturbation for 100 different mainmagnetic field orientations according to the von-Mises-Fisher distribution [58], the final signal is the sum of signalfrom 2D models with the 100 different orientations inrespect to the main magnetic field.

The 3D models were compared to 10 realistic 2D mod-els, created as described in the methods section, usingsimilar microstructural parameters to those of the 3Dsamples. Four different different dispersion values (σ =0, 0.2, 0.4, 0.6) were simulated. The ME-GRE signals werecomputed for both 2D and 3D models, with the parameterused in Fig I for TE = 1:1:80 ms. Finally, the 2D and3D signals were normalized and compared using the root-mean-square-error (RMSE) computed according to:

RMSE(S3D, S2D) =

√< (S3D − S2D), (S3D − S2D) >

#TE(7)

where <.,.> is the complex dot product and #TE is thenumber of echos.

Figure 12 shows the signal RMSE between the 2D and3D models as a function of the orientation of the mainmagnetic field. In each plot various 2D simulated signalswith different dispersion levels are compared to (a) theoriginal 3D model (b) the 3D model with high dispersion.The 2D models with lower dispersion (0 0.2) consistentlymatch that 3D signal with RMSEs below the 2.5%, whichis small when taking into account the 4% noise added tothe training of our deep learning network. For the highdispersion 3D model (Fig. 12b), the 2D models with highdispersion (0.4 and 0.6) have the lowest RMSE for allmagnetic field orientations. When no dispersion is usedin the 2D models, the RMSE stays below 5%. The two 3Dmodels considered are best represented with 2D modelswith similar or slightly higher dispersion values. Thisfinding could be attributed to the additional dispersionassociated with each axon that changes direction through-out the 3D model and that is not taken into account inthe current dispersion computation.

To conclude, the developed 2D models based on separatelibrary of axons accurately represent a real 3D whitematter model. In the future, it could be considered to add

Fig. 12. Plots of the RMSE between the Signal of the 2D modelsusing 4 different dispersion levels and Signal of the 3D models asa function of the orientation of the main magnetic field. In a) theoriginal 3D model with low dispersion (0.04) and in b) the 3D modelwith high dispersion (0.4)is used as ground truth. The error barsrepresent the standard deviation across 1o different realistic 2D WMmodels created

dispersion to the 2D models to better represent a whitematter regions with higher dispersion value that could bemeasured independently with DWI. In ex vivo acquisitionsthe quality of DTI data is severely hampered (reduceddiffusion constant and reduced T2) and from our data itwas not possible to apply more advanced diffusion modelsthat can decode this quantity. However, even withoutdispersion, the RMSE consistently satayed under 5% while4% noise is added to our dictionary when traing the deeplearning network, which suggests that this might not havea large impact.

A situation not considered here and that should havea larger impact are crossing fibers. Fiber dispersion, dis-cussed above, accounts for the spread of the fiber orien-tations within a bundle of axons while the fiber crossingrepresents two or more bundles of axons. Significant workon the diffusion community has been devoted to this topic[59]. This could be studied as a future work assuming thatsuch a 3D EM dataset exists.

References

[1] B. Zalc, “The acquisition of myelin: a success story,” in NovartisFoundation symposium, vol. 276. Wiley Online Library, 2006,p. 15.

[2] S. Love, “Demyelinating diseases,” Journal of clinical pathology,vol. 59, no. 11, pp. 1151–1159, 2006.

[3] G. Bonnier, B. Marechal, M. J. Fartaria, P. Falkowskiy, J. P.Marques, S. Simioni, M. Schluep, R. Du Pasquier, J.-P. Thiran,G. Krueger et al., “The combined quantification and inter-pretation of multiple quantitative magnetic resonance imagingmetrics enlightens longitudinal changes compatible with brainrepair in relapsing-remitting multiple sclerosis patients,” Fron-tiers in neurology, vol. 8, p. 506, 2017.

[4] J. Du, V. Sheth, Q. He, M. Carl, J. Chen, J. Corey-Bloom,and G. M. Bydder, “Measurement of t1 of the ultrashort t2*components in white matter of the brain at 3t,” PloS one, vol. 9,no. 8, 2014.

[5] Q. He, Y. Ma, S. Fan, H. Shao, V. Sheth, G. M. Bydder, andJ. Du, “Direct magnitude and phase imaging of myelin usingultrashort echo time (ute) pulse sequences: A feasibility study,”Magnetic resonance imaging, vol. 39, pp. 194–199, 2017.

[6] S. D. Wolff and R. S. Balaban, “Magnetization transfer contrast(mtc) and tissue water proton relaxation in vivo,” Magneticresonance in medicine, vol. 10, no. 1, pp. 135–144, 1989.

[7] J. G. Sled, “Modelling and interpretation of magnetizationtransfer imaging in the brain,” Neuroimage, vol. 182, pp. 128–135, 2018.


https://doi.org/10.1101/2020.06.23.127258

15

[8] E. Alonso-Ortiz, I. R. Levesque, and G. B. Pike, “Multi-gradient-echo myelin water fraction imaging: Comparison tothe multi-echo-spin-echo technique,” Magnetic resonance inmedicine, vol. 79, no. 3, pp. 1439–1446, 2018.

[9] K. Susuki, “Myelin: a specialized membrane for cell communi-cation,” Nature Education, vol. 3, no. 9, p. 59, 2010.

[10] J. H. Duyn and J. Schenck, “Contributions to magnetic suscep-tibility of brain tissue,” NMR in Biomedicine, vol. 30, no. 4, p.e3546, 2017.

[11] T. Xu, S. Foxley, M. Kleinnijenhuis, W. C. Chen, and K. L.Miller, “The effect of realistic geometries on the susceptibility-weighted mr signal in white matter,” Magnetic resonance inmedicine, vol. 79, no. 1, pp. 489–500, 2018.

[12] D. A. Yablonskiy and A. L. Sukstanskii, “Biophysical mech-anisms of myelin-induced water frequency shifts,” Magneticresonance in medicine, vol. 71, no. 6, p. 1956, 2014.

[13] P. Sati, P. van Gelderen, A. C. Silva, D. S. Reich, H. Merkle,J. A. De Zwart, and J. H. Duyn, “Micro-compartment specifict2a ↪AO relaxation in the brain,” Neuroimage, vol. 77, pp. 268–278, 2013.

[14] S. Wharton and R. Bowtell, “Fiber orientation-dependent whitematter contrast in gradient echo mri,” Proceedings of the Na-tional Academy of Sciences, vol. 109, no. 45, pp. 18 559–18 564,2012.

[15] T. Xu, “Biophysical modeling of white matter in magneticresonance imaging,” Ph.D. dissertation, University of Oxford,2017.

[16] W. Jung, S. Bollmann, and J. Lee, “Overview of quantita-tive susceptibility mapping using deep learning–current status,challenges and opportunities,”arXiv preprint arXiv:1912.05410,2019.

[17] J. Lee, K. Shmueli, B.-T. Kang, B. Yao, M. Fukunaga,P. Van Gelderen, S. Palumbo, F. Bosetti, A. C. Silva, and J. H.Duyn, “The contribution of myelin to magnetic susceptibility-weighted contrasts in high-field mri of the brain,” Neuroimage,vol. 59, no. 4, pp. 3967–3975, 2012.

[18] R. Gil, D. Khabipova, M. Zwiers, T. Hilbert, T. Kober, and J. P.Marques, “An in vivo study of the orientation-dependent andindependent components of transverse relaxation rates in whitematter,” NMR in Biomedicine, vol. 29, no. 12, pp. 1780–1790,2016.

[19] J. Lee, H.-G. Shin, W. Jung, Y. Nam, S.-H. Oh, and J. Lee,“An r2* model of white matter for fiber orientation and myelinconcentration,” Neuroimage, vol. 162, pp. 269–275, 2017.

[20] S. Papazoglou, T. Streubel, M. Ashtarayeh, K. J. Pine, L. J.Edwards, M. Brammerloh, E. Kirilina, M. Morawski, C. Jager,S. Geyer et al., “Biophysically motivated efficient estimation ofthe spatially isotropic component from a single gradient-recalledecho measurement,” Magnetic resonance in medicine, vol. 82,no. 5, pp. 1804–1811, 2019.

[21] Y. Nam, J. Lee, D. Hwang, and D.-H. Kim, “Improved esti-mation of myelin water fraction using complex model fitting,”NeuroImage, vol. 116, pp. 214–221, 2015.

[22] P. Van Gelderen, J. A. De Zwart, J. Lee, P. Sati, D. S. Reich,and J. H. Duyn, “Nonexponential t2* decay in white matter,”Magnetic resonance in medicine, vol. 67, no. 1, pp. 110–117,2012.

[23] E. Alonso-Ortiz, I. R. Levesque, and G. B. Pike, “Impact ofmagnetic susceptibility anisotropy at 3 t and 7 t on t2*-basedmyelin water fraction imaging,” NeuroImage, vol. 182, pp. 370–378, 2018.

[24] J. Lee, Y. Nam, J. Y. Choi, E. Y. Kim, S.-H. Oh, and D.-H.Kim, “Mechanisms of t2* anisotropy and gradient echo myelinwater imaging,” NMR in Biomedicine, vol. 30, no. 4, p. e3513,2017.

[25] S. Wharton and R. Bowtell, “Gradient echo based fiber orienta-tion mapping using r2* and frequency difference measurements,”Neuroimage, vol. 83, pp. 1011–1023, 2013.

[26] K.-S. Chan and J. P. Marques,“Brain tissue multi-compartmentrelaxometry - an improved method for in vivo myelin waterimaging,” in ISMRM, 2019.

[27] A. Zaimi, T. Duval, A. Gasecka, D. Cote, N. Stikov, andJ. Cohen-Adad, “Axonseg: open source software for axon andmyelin segmentation and morphometric analysis,” Frontiers inneuroinformatics, vol. 10, p. 37, 2016.

[28] T. Mingasson, T. Duval, N. Stikov, and J. Cohen-Adad, “Axon-packing: an open-source software to simulate arrangements ofaxons in white matter,” Frontiers in neuroinformatics, vol. 11,p. 5, 2017.

[29] F. Sepehrband, K. A. Clark, J. F. Ullmann, N. D. Kurni-awan, G. Leanage, D. C. Reutens, and Z. Yang, “Brain tissuecompartment density estimated using diffusion-weighted mriyields tissue parameters consistent with histology,”Human brainmapping, vol. 36, no. 9, pp. 3687–3702, 2015.

[30] W. Li, C. Liu, T. Q. Duong, P. C. van Zijl, and X. Li, “Suscep-tibility tensor imaging (sti) of the brain,” NMR in Biomedicine,vol. 30, no. 4, p. e3540, 2017.

[31] S. W. Choy, E. Bagarinao, H. Watanabe, E. T. W. Ho, S. Mae-sawa, D. Mori, K. Hara, K. Kawabata, N. Yoneyama, R. Ohdakeet al., “Changes in white matter fiber density and morphologyacross the adult lifespan: A cross-sectional fixel-based analysis,”Human Brain Mapping, 2020.

[32] S. Mohammadi, D. Carey, F. Dick, J. Diedrichsen, M. I. Sereno,M. Reisert, M. F. Callaghan, and N. Weiskopf, “Whole-brain in-vivo measurements of the axonal g-ratio in a group of 37 healthyvolunteers,” Frontiers in neuroscience, vol. 9, p. 441, 2015.

[33] J. Mollink, M. Kleinnijenhuis, A.-M. v. C. van Walsum, S. N.Sotiropoulos, M. Cottaar, C. Mirfin, M. P. Heinrich, M. Jenkin-son, M. Pallebage-Gamarallage, O. Ansorge et al., “Evaluatingfibre orientation dispersion in white matter: Comparison of dif-fusion mri, histology and polarized light imaging,” Neuroimage,vol. 157, pp. 561–574, 2017.

[34] K. Ginsburger, F. Poupon, J. Beaujoin, D. Estournet, F. Ma-tuschke, J.-F. Mangin, M. Axer, and C. Poupon, “Improvingthe realism of white matter numerical phantoms: a step towarda better understanding of the influence of structural disordersin diffusion mri,” Frontiers in Physics, vol. 6, p. 12, 2018.

[35] D. Ma, V. Gulani, N. Seiberlich, K. Liu, J. L. Sunshine, J. L.Duerk, and M. A. Griswold, “Magnetic resonance fingerprint-ing,” Nature, vol. 495, no. 7440, pp. 187–192, 2013.

[36] F. Chollet et al., “Keras,” https://keras.io, 2015.[37] M. Jenkinson, P. Bannister, M. Brady, and S. Smith, “Improved

optimization for the robust and accurate linear registration andmotion correction of brain images,” Neuroimage, vol. 17, no. 2,pp. 825–841, 2002.

[38] V. Arsigny, P. Fillard, X. Pennec, and N. Ayache,“Log-euclideanmetrics for fast and simple calculus on diffusion tensors,” Mag-netic Resonance in Medicine: An Official Journal of the Inter-national Society for Magnetic Resonance in Medicine, vol. 56,no. 2, pp. 411–421, 2006.

[39] X. Li, D. S. Vikram, I. A. L. Lim, C. K. Jones, J. A. Farrell, andP. C. van Zijl, “Mapping magnetic susceptibility anisotropies ofwhite matter in vivo in the human brain at 7 t,” Neuroimage,vol. 62, no. 1, pp. 314–330, 2012.

[40] D. A. Yablonskiy, “Quantitation of intrinsic magneticsusceptibility-related effects in a tissue matrix. phantomstudy,” Magnetic resonance in medicine, vol. 39, no. 3, pp.417–428, 1998.

[41] D. A. Yablonskiy and A. L. Sukstanskii, “Lorentzian effectsin magnetic susceptibility mapping of anisotropic biologicaltissues,” Journal of Magnetic Resonance, vol. 292, pp. 129–136,2018.

[42] P. van Gelderen and J. H. Duyn, “White matter intercompart-mental water exchange rates determined from detailed modelingof the myelin sheath,” Magnetic resonance in medicine, vol. 81,no. 1, pp. 628–638, 2019.

[43] A. L. Sukstanskii, J. Wen, A. H. Cross, and D. A. Yablonskiy,“Simultaneous multi-angular relaxometry of tissue with mri(smart mri): Theoretical background and proof of concept,”Magnetic resonance in medicine, vol. 77, no. 3, pp. 1296–1306,2017.

[44] D. J. Pinero and J. R. Connor, “Iron in the brain: an importantcontributor in normal and diseased states,” The Neuroscientist,vol. 6, no. 6, pp. 435–453, 2000.

[45] W. Li, B. Wu, A. V. Avram, and C. Liu,“Magnetic susceptibilityanisotropy of human brain in vivo and its molecular underpin-nings,” Neuroimage, vol. 59, no. 3, pp. 2088–2097, 2012.

[46] A. S. Shatil, M. N. Uddin, K. M. Matsuda, and C. R. Figley,“Quantitative ex vivo mri changes due to progressive formalinfixation in whole human brain specimens: longitudinal charac-terization of diffusion, relaxometry, and myelin water fractionmeasurements at 3t,” Frontiers in medicine, vol. 5, p. 31, 2018.


https://doi.org/10.1101/2020.06.23.127258

16

[47] C. Birkl, M. Soellradl, A. M. Toeglhofer, S. Krassnig, M. Leoni,L. Pirpamer, T. Vorauer, H. Krenn, J. Haybaeck, F. Fazekaset al., “Effects of concentration and vendor specific compositionof formalin on postmortem mri of the human brain,” Magneticresonance in medicine, vol. 79, no. 2, pp. 1111–1115, 2018.

[48] T. M. Shepherd, J. J. Flint, P. E. Thelwall, G. J. Stanisz,T. H. Mareci, A. T. Yachnis, and S. J. Blackband, “Postmorteminterval alters the water relaxation and diffusion properties ofrat nervous tissueaATimplications for mri studies of humanautopsy samples,”Neuroimage, vol. 44, no. 3, pp. 820–826, 2009.

[49] R. Sener, “Diffusion mri: apparent diffusion coefficient (adc)values in the normal brain and a classification of brain disor-ders based on adc values,” Computerized medical imaging andgraphics, vol. 25, no. 4, pp. 299–326, 2001.

[50] F. A. Duck, Physical properties of tissues: a comprehensivereference book. Academic press, 2013.

[51] L. E. Lee, E. Ljungberg, D. Shin, C. R. Figley, I. M. Vavasour,A. Rauscher, J. Cohen-Adad, D. K. Li, A. L. Traboulsee, A. L.MacKay et al., “Inter-vendor reproducibility of myelin waterimaging using a 3d gradient and spin echo sequence,” Frontiersin neuroscience, vol. 12, p. 854, 2018.

[52] K.-S. Chan and J. P. Marques, “Diffusion-weighted imaginginformed gradient-echo myelin water imaging,” in ESMRMB,2019.

[53] M. Morawski, E. Kirilina, N. Scherf, C. Jager, K. Reimann,R. Trampel, F. Gavriilidis, S. Geyer, B. Biedermann, T. Arendtet al., “Developing 3d microscopy with clarity on human braintissue: Towards a tool for informing and validating mri-basedhistology,” Neuroimage, vol. 182, pp. 417–428, 2018.

[54] S. R. Stock, Microcomputed tomography: methodology and ap-plications. CRC press, 2019.

[55] N. De Jonge, R. Sougrat, B. M. Northan, and S. J. Pennycook,“Three-dimensional scanning transmission electron microscopyof biological specimens,” Microscopy and Microanalysis, vol. 16,no. 1, pp. 54–63, 2010.

[56] M. Kleinnijenhuis et al., “A 3d electron microscopy segmenta-tion pipeline for hyper-realistic diffusion simulations,” 2017.

[57] (2019) Fsl - dispersion calculation. AnalysisGroup, FMRIB, Oxford, UK. [Online]. Avail-able: https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FDT/UserGuide?action=AttachFile&do=view&target=dyads.jpg

[58] A. Banerjee, I. S. Dhillon, J. Ghosh, and S. Sra, “Clusteringon the unit hypersphere using von mises-fisher distributions,”Journal of Machine Learning Research, vol. 6, no. Sep, pp. 1345–1382, 2005.

[59] H. Farooq, J. Xu, J. W. Nam, D. F. Keefe, E. Yacoub, T. Geor-giou, and C. Lenglet, “Microstructure imaging of crossing (mix)white matter fibers from diffusion mri,” Scientific reports, vol. 6,p. 38927, 2016.


https://doi.org/10.1101/2020.06.23.127258

white matter microstructural property decoding from gradient echo data using realistic ... ·...

Documents